Scientific Papers - Vi

394- ON LEGENDRE'S FUNCTION Pn (Q), [404
If we take out a further factor, eime, writing
u = vsm~^0 = weimesm~^6, .....................(3)
of which ultimately only the real part is to be retained, we find
dhu . dw w ,.v
d& + 2mldd + fStf-e = °.........................(4)
We next change the independent variable to z> equal to cot 6, thus
obtaining
dw -i (,_ d*w diu w
-7-= 5— u cte 2m (v
From this equation we can approximate to the desired solution, treating m as a large quantity and supposing that w = \ when 0 — 0, or $ = ^TT.
The second approximation gives
dw i , ., *'# •
_ _ — whence w = l— -5— . rf5 8m 8m
After two more steps we find
~ i • f JL _ 9_J _ 9-g2 , 75^a p,
W~ ^V8m 128mV 128m2+1024ms.............l ;
Thus in realized form a solution of (1) is
^^)oo8(«K9 + 7)
foot 5 9cot'0 75 cot3 0) . , Q '] /17N
+ ^-5--------T-JJ5—- — T^oT—rh sin (me' -f 7) ; ......(7)
(8m 128m3 1024m3] • ' _]
and this may be identified with Pn provided that the constants C, 7, can be so chosen that u and du/dd have the correct values when 0 = ^7r. For this value of ^ we must have
-Pn(i7r) = (7cos(£m7r + 7),........................(8)
(dPn/dd)^ = (7 (- m - §L + __!_) Sin (im?r + 7)..........(9)
We may express (dPnld6\ by means of Pn+1 (^-TT). In general
rfP
Sin2 ^ r7 ^
w LUo (7
so that when 0 = £ TT,
dPn/de = ~dPn/dcQBd = (n + l)Pn+l................(10)
When wis even,(dPn/dd)^v vanishes, and, G being still undetermined, we may take to satisfy (9), 7 = - |TT ; and then from (8)
(7 cos (Jn*-) = Pw (i,r) = (- I)*'1 •"^"^ ,n as equal to unity. W. F. S.] unexpectedly, that there is a linear acceleration amounting to